Approximate Counting of Graphical Realizations

Erdős, Péter L.; Kiss, Sándor Z.; Miklós, István; Soukup, Lajos

doi:10.1371/journal.pone.0131300

Cited by 20 publications

(40 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The meaning of almost half-regular is analogous. (F) was proved by Erdős, Kiss, Miklós and Soukup [22]. The process uses C 4 -and C 6 -swaps, so while it contains the directed degree sequence problem as a special case, it is not comparable with the result in Theorem 2.1(B).…”

Section: Markov Chain Monte Carlo Samplingmentioning

confidence: 97%

“…forbidden (u x , v x ) type edges. This problem class was introduced in paper [22], along with a Havel-Hakimi-type graphicality test for restricted bipartite degree sequences.…”

Section: Graphsmentioning

confidence: 99%

“…Here the degree sequence on one side of the partition is regular, while the degrees can be arbitrary on the other side. Most recently, Greenhill proved the conjecture for simple graphs with relatively small maximal degrees, and also recently, Erdős, Kiss, Miklós and Soukup [22] proved the conjecture for almost half-regular bipartite graphs with certain forbidden edge-sets. (Comprehensive surveys of the topic can be found in [27] and [40].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

New Classes of Degree Sequences with Fast Mixing Swap Markov Chain Sampling

Erdős¹,

Miklós

Toroczkai³

2017

Combinator. Probab. Comp.

Self Cite

View full text Add to dashboard Cite

One of the simplest methods of generating a random graph with a given degree sequence is provided by the Monte Carlo Markov Chain method using swaps. The swap Markov chain converges to the uniform distribution, but generally it is not known whether this convergence is rapid or not. After a number of results concerning various degree sequences, rapid mixing was established for so-called P -stable degree sequences (including that of directed graphs), which covers every previously known rapidly mixing region of degree sequences.In this paper we give a non-trivial family of degree sequences that are not P -stable and the swap Markov chain is still rapidly mixing on them. The proof uses the 'canonical paths' method of Jerrum and Sinclair. The limitations of our proof are not fully explored.

show abstract

Section: Markov Chain Monte Carlo Samplingmentioning

confidence: 97%

“…forbidden (u x , v x ) type edges. This problem class was introduced in paper [22], along with a Havel-Hakimi-type graphicality test for restricted bipartite degree sequences.…”

Section: Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

New Classes of Degree Sequences with Fast Mixing Swap Markov Chain Sampling

Erdős¹,

Miklós

Toroczkai³

2017

Combinator. Probab. Comp.

Self Cite

View full text Add to dashboard Cite

show abstract

“…However, any argument which gives an upper bound on |L * (Z)|/|Ω(d)| will do. So we will instead consider a slightly more complicated operation than a switch, which we call a 3-switch (this operation is called a "circular C 6 -swap" in [10]). (This approach of considering more complicated operations in order to obtain more freedom has been used to improve asymptotic enumeration results, for example in [30].)…”

Section: This Proves (I)mentioning

confidence: 99%

The switch Markov chain for sampling irregular graphs and digraphs

Greenhill

Sfragara

2018

Theoretical Computer Science

107

View full text Add to dashboard Cite

The problem of efficiently sampling from a set of (undirected, or directed) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the sampling. The switch chain is known to be rapidly mixing for regular degree sequences, both in the undirected and directed setting.We prove that the switch chain for undirected graphs is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree d max which satisfies 3 ≤ d max ≤ 1 3 √ M , where M is the sum of the degrees. The mixing time bound obtained is only a factor n larger than that established in the regular case, where n is the number of vertices. Our result covers a wide range of degree sequences, including power-law density-bounded graphs with parameter γ > 5/2 and sufficiently many edges.For directed degree sequences such that the switch chain is irreducible, we prove that the switch chain is rapidly mixing when all in-degrees and out-degrees are positive and bounded above by 1 4 √ m, where m is the number of arcs, and not all in-degrees and out-degrees equal 1. The mixing time bound obtained in the directed case is an order of m 2 larger than that established in the regular case.Here the running time of the sampling algorithm must be (deterministically) polynomially bounded but the output need not be exactly uniform: however, the user can specify how far from the uniform distribution the samples may be. Other approaches to the problem of sampling graphs (or directed graphs) are discussed in Section 1.1.The switch chain is a natural and well-studied Markov chain for sampling from a set of graphs with a given degree sequence. Each move of the switch chain selects two distinct edges uniformly at random and attempts to replace these edges by a perfect matching of the four endvertices, chosen uniformly at random. The proposed move is rejected if the four endvertices are not distinct or if a multiple edge would be formed. We call each such move a switch. The precise definitions of the transitions for the switch chain for undirected and directed graphs are given at the start of Sections 2 and 3, respectively.Ryser [34] used switches to study the structure of 0-1 matrices. Markov chains based on switches have been introduced by Besag and Clifford [5] for 0-1 matrices (bipartite graphs), Diaconis and Sturmfels [9] for contingency tables and Rao, Jana and Bandyopadhyay [33] for directed graphs.The switch chain is aperiodic and its transition matrix is symmetric. It is well-known that the switch chain is irreducible for any (undirected) degree sequence: see [32,37]. Irreducibility for the directed chain is not guaranteed, see Rao et al. [33]. However, Berger and Müller-Hanneman [4] and LaMar [24,26] gave characterisations of directed degree sequences for which the switch chain is irreducible. In particular, the switch chain is irreducible for regular directed graphs (see for example Greenhill [15, Lemma 2.2]).In order for the switch chain to b...

show abstract

“…It seems unlikely that the approach of Cooper et al [4] can be adapted to the setting of sampling d-factors, due to the complexity of the analysis. Erdős et al [6] analysed a Markov chain algorithm which uniformly generates bipartite graphs with a given half-regular degree sequence, avoiding a set of edges which is the union of a 1-factor and a star. Here "half-regular" means that the degrees on one side of the bipartition are all the same, with the possible exception of the centre of the star.…”

Section: Introductionmentioning

confidence: 99%

Uniform Generation of Spanning Regular Subgraphs of a Dense Graph

Gao

Greenhill²

2019

Electron. J. Combin.

View full text Add to dashboard Cite

Let H n be a graph on n vertices and let H n denote the complement of H n . Suppose that ∆ = ∆(n) is the maximum degree of H n . We analyse three algorithms for sampling d-regular subgraphs (d-factors) of H n . This is equivalent to uniformly sampling dregular graphs which avoid a set E(H n ) of forbidden edges. Here d = d(n) is a positive integer which may depend on n.Two of these algorithms produce a uniformly random d-factor of H n in expected runtime which is linear in n and low-degree polynomial in d and ∆. The first algorithm applies when (d + ∆)d∆ = o(n). This improves on an earlier algorithm by the first author, which required constant d and at most a linear number of edges in H n . The second algorithm applies when H n is regular and d 2 + ∆ 2 = o(n), adapting an approach developed by the first author together with Wormald. The third algorithm is a simplification of the second, and produces an approximately uniform d-factor of H n in time O(dn). Here the output distribution differs from uniform by o(1) in total variation distance, provided that d 2 + ∆ 2 = o(n). d = o(n 1/3 ), and by McKay and Wormald [16] to d = o( √ n), for the d-regular graph case.On the other hand, asymptotic enumeration for dense d-regular graphs was performed in 1990 by McKay and Wormald [17] for d, n − d ≥ n/ log n, leaving a gap in which no result was known: namely, when min{d, n − d} lies between n 1/2 and n/ log n. This gap was filled recently by Liebenau and Wormald [13]. Uniform generation of graphs with given degrees has an equally long history, and is closely related to asymptotic enumeration. Tinhofer [19] is among the first who investigated algorithmic heuristics, and realised that efficient uniform generation, or approximately uniform generation, is difficult. Graph enumeration arguments can sometimes be adapted to provide algorithms for uniform generation of graphs with given degrees. The proofs in [2,3] immediately yield a simple rejection algorithm, which uniformly generates graphs of degrees at most O( √ log n) in expected polynomial time (polynomial as a function of n). The switching arguments in [14,16] were adapted to give a polynomial-time (in expectation) algorithm [15] for uniformly generating random d-regular graphs when d = O(n 1/3 ). However, overcoming the n 1/3 barrier was extremely challenging, with no progress for more than two decades. A major breakthrough was obtained by Wormald and the first author [8,9], by modifying the McKay-Wormald algorithm [15] to a more flexible form, which allows the use and classification of different types and classes of switchings. This new technique greatly extended the range of degree sequences for which uniform generation is possible. The first application [8,9] of the new technique gave an algorithm for uniform generation of d-regular graphs with expected runtime O(nd 3 ) when d = o( √ n). The second application [10] gave an algorithm for uniform generation of graphs with power-law degree sequences with exponent slightly below 3, with expected runtime O(n 2.107 ) with...

show abstract

Approximate Counting of Graphical Realizations

Cited by 20 publications

References 22 publications

New Classes of Degree Sequences with Fast Mixing Swap Markov Chain Sampling

New Classes of Degree Sequences with Fast Mixing Swap Markov Chain Sampling

The switch Markov chain for sampling irregular graphs and digraphs

Uniform Generation of Spanning Regular Subgraphs of a Dense Graph

Contact Info

Product

Resources

About